3.8. CARRIER GENERATION AND RECOMBINATION 135
whereEis thermal energy, 3 / 2 kBTin three dimensions. Thus
vth=√
3 kBT
m∗
107 cm/sec (3.8.28)For the electron emission process, b,
rb=enNtf (3.8.29)
whereenis the emission rate from the trap andNtfis the concentration of occupied traps. The
capture rate for holes, process c, will be analogous to process a with the difference that holes are
captured by occupied traps.
rc=vthσppNtf
Finally, the emission of holes has a rate:
rd=epNt(1−f) (3.8.30)whereepis the emission probability for holes. The next step is to determine the emission proba-
bilitiesenandep. In general this is a very difficult problem since,fis known only in equilibrium.
So first consider the equilibrium values ofen,andep. In equilibrium transition rates into and out
of the conduction band must be equal, orra=rb. Inserting
n=Ncexp (−(EC−EF)/kBT)=niexp ((EF−Ei)/kBT) (3.8.31)intora=rbleads to:
en=vthσnniexp ((Et−Ei)/kBT) (3.8.32)or
en=vthσnNCexp (−(EC−Et)/kBT) (3.8.33)
Thus the emission probability of electrons into the conduction band rises exponentially as the
trap gets closer toECwhich we expect intuitively. Fromrc=rdand
p=NVexp [−(Ef−EV)/kBT]ep=vthσpNV exp (−(EV−Et)/kBT)
=vthσpniexp (+ (Ei−Et)/kBT)In non-equilibrium (the case of most interest)fis unknown and has to be calculated. To do
so, rate equations are solved. Assume that non-equilibrium is generated by optical excitation
resulting in a generation rate ofGLelectron-hole pairs/second. We also assume that the emission
rates,en,andepare not a function of illumination and the same as that calculated at equilibrium.
In steady state, the concentration of electrons,nnand holes,pnin ann−type semiconductor
is not a function of time and from figure 3.25 we get:
dnn
dt=GL−(ra−rb)=0 (3.8.34)
dPn
dt=GL−(rc−rd)=0 (3.8.35)
∴ra−rb=rc−rd (3.8.36)